Sahai&Emi
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Sahai&Emi

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FLOW

vp/V e=>

Vm

'

Vp/V o•->

(b)

fLllIPr"~Ul'~~Us_rEc INPUT MIXEDFLOW
o e--~

Vm

c
PLUGFLOW

o e•-~
VP

c

v !V e'-~'
Fig, l.

Acombinedmodel representing plug volume and
mixed volume.

C ::-Lmax Vm
1,
C

[- )J

v"'p vl~
(e-l~v_'vl Q=Q ACTIVEREGION

Va

Omin
=

=y Time
,
eV

Fig. 2. C-curve for a combinedmodel presented in Fig, l.

DEAD
REGION Qd= O

Vd

Frg. 3. Flow through active and dead (stagnant) regions of a
combinedmodel.

* Meanresidence time is the ratio of the volume of liquid to the volumetric flow rate.

@1996 ISIJ 668

ISIJ International, Vol. 36 (1996), No. 6

Q
Qa

ACTIVE
REGION

V,

Qd DEAD
REGION

Vd

Q

ACTIVE QREGION
Va

Qd
DEAD
REGION

Vd

Flow through active and dead (slow moving) regions
of a combinedmodel.

C
Q

1.2

0.8

0.6

0.4

0.2

O

Total Area up to e= 2
Q*
Q

Area Q-dQ

Fig. 4.

in the active region. Thus, the fluid which stays in the
vessel for a period longer than two times the meanresi-
dence time is considered as the dead volume. Twoalter-
native waysof schematically representing a system with
slow movingdead volume are shownin Fig. 4. Thedead
volurne in most of the normally operating tundishes falls
in the second type, and is characterized by a long tail
extending beyondthe two times the meanresidence time.

The average residence time of the fluid for any given
tundish at a constant volumetric flow rate remains con-
stant. Thus, the slower movingfluid or deadvolumestays
10nger in the tundish at the expense of other fluid. In
other words, if somefluid assumesmuchlonger residence
time in the tundish, an equivalent amountof other fluid
has, accordingly, a shorter residence time in the tundish.
This faster moving melt maynot spend sufficient time
to separate and fioat out the non-metallic inclusions.
Also, molten metal in the dead (slow moving) regions
mayloose sufficient heat, andmaystart to solidify metal.
Thus, tundishes are designed to have dead volume as
small as possible.

Consider a combined model consisting of an active
(plug fiow and well-mixed fiow) and a dead regions. As
depicted in Figs. 3and 4, Iet the total volume of the
system be Vwhich is divided into an active volume of
V* and a dead volume of Vd. Let the total volumetric
flow rate through the system be Qwhich is also divided
in Q* through the active region and Qdthrough the dead
region. For completely stagnant deadvolume (represent-
ed in Fig. 3). QdWill be zero and Qwill be equal to Q*.

For a dead region with slowly moving fluid, a typical
experimentally obtained RTDis shownin Fig. 5. ARTD
curve corresponding to a pulse input is knownas the
C-curve. Let the dimensionless meantime of the C-curve
upto the cutoff point of dimensionless time, O=2be ~c,then

e~c
=

measuredmeantime upto e=2 measuredt~c
meanresidence time ~t

.(1)

O 1 2 43
~.=Jy

.~ e
v Q.

Fig. 5. A typical residence time distribution curve for flow in
a tundish

e~c
tc V./Qa

_

V.
.

Q ,.......,.(2)
~ ~ V/Q ~ V Qa """~

Va
_

Qa
. ~

..........(3)V~Q c
Thus, the dead volume fraction

Vd I Qa . O~c """""(4)V Q
The term Q./Q is the area under the C-curve from e=0
to 2, and represents the fractional volumetric flow rate
through the active region. With the presence of dead
region(s), the measuredaverage dimensionless residence
time,

ec
......

..........(5)

If the dead region is completely stagnant (as in Fig.
3) so that the fiowing fluid does not enter or leave the
region, the volumeof the system through which the fluid
flows in the system is effectively reduced to Va (or QalQ
is one in Eq. (4)). Thus, the deadvolume fraction will be

V
= I -ec """""(6)

The dead volume fraction with stagnant volume is
given by Eq. (6), which is a special case of Eq. (4). The
dead volume with the slowly moving fluid is given by
Eq. (4).

5. Application of the CombinedModel to Melt Flow in
Tundishes

As stated earlier, a typical experimental C-curve ob-
tained in water model studies or in an actual tundish
showsan extended tail beyond the time, 0=2. This in-
dicates the existence of a slow moving flow through the
dead regions. As shownschematically in Fig. 6, there
mayexist dead regions on the downstreamside of the
damsand weirs, or near the end wall. Figures 7and 8
are taken from literatures where detailed flow patterns
in different tundishes are predicted by the solution of
the Navier Stokes' equation. Figure 7is taken from the

669 C 1996 Is[J

ISIJ International, Vol. 36 (1996), No. 6

DeadVolume

.\~..x\",r*
"'//.

'//..

Table l. Estimated error in dead volume calculation by
ignoring the QdlQ.

Qd/Q Correct Vd/V (Eq. (4)) Elrol In V IV

O
0.01
0.05
O. 1

O, l
O, I09
O, 145
0,19

O
90/0

450/0
900/0

Frg. 6. Representation of melt flow in a tundish having dead
volumeand exchangeof liquid with the active volume.

Inlet End

O.OS M/S
=-~>

~
\

-- L
'/'L'~~._*.

~'
li

L 1"'I-ff'-!T
-

"
' : i:~

.,~~

l\
,,('

Continuous temperature measl

l

/ ~ / /////
~/ ///~/ / / 't

~//// /~ /ll/
, ,~

~
); (/// \~~

~// ~/~/'7/__
_ _

-"~~:P(///_
___'~~/// '

_

-
/ - //' ~_

\
~

~/ f// ,Channel /;;
Strand

\\\- - /
Fig. 7. Predicted low pattern in a selected plane ofthe Armco

KansasCity bloom caster tundish (Ref. 2)).

measurement

Fig. 8. Predicted velocity profile in an induction heated tun-
dish at the Muroran works of the Nippon Steel Cor-
poration (Ref. 3)).

work of Lowry and Sahai2) and showsflow pattern in a
selected plane of the ArmcoKansasCity bloom caster
tundish. The figure clearly depicts the slow movingdead
volumes behind the damand weir. Figure 8shows ve-
locity profile predicted by Suzuki et al.3) in an induc-
tion heated tundish at the Muroranworks of the Nippon
Steel Corporation. In these figures, the velocity vectors
are proportional to the magnitude of velocity at that
Iocation. These figures show regions of very high ve-
locity and thus, high turbulence and regions of good
mixing, and regions with very slow moving fluid causing
dead volumes.

The fiuid in these dead regions is constantly inter-
changedwith the main flow (in active volume) of the tun-
dish. Theseregions should not be considered as stagnant
deadvolume regions. Thus, the deadvolume fraction in
tundishes is given by Eq. (4). There are two approaches
used by researchers in the analysis and modeling of the
melt flow in tundishes. The first one, which has been
most widely used (e.g. Refs. 4)-6)), is the use of Eq. (6).
Thus, the model assumesthat the area under the curve

@1996 ISIJ 670

from the time, 0=2 to co in Fig. 5is zero. This area rep-
resents the fraction of the volumetric flow rate through
the dead regions (QdlQ)• Onecan visualize in any water
modeling experiments of a typical tundish design that
there is always an exchangeof liquid between the main
flow (active volume) and the so called dead voiume re-
gions. This assumption maylead to someerror in the
ca]culation.

In the secondapproach (e.g. Ref. 7)), the deadvolume
fraction has been considered to be equal to the area
under the curve from the time 0=2 to co. Both ap-
proaches normally lead to incorrect determination of the
dead volume. Table I gives someestimates of error in
the dead volume calculation by using Eq. (6). In these
calculations, it is assumedthat the deadvolumewithout
any flow through the dead region is lO"/o of the total
volume. With the fractional flow through the dead re-
gions (Qd/Q) of l, 5, and 10 */o, the correct dead volume
calculated from Eq. (4) and the error by using Eq. (6)
are given in the table. It can be seen that the error may
be as high as 90 o/o with the 10 o/. exchangeor cross flow
between the dead and active regions.

6. Calculation of the Plug and Mixed Volumes
After calculation of the dead volume, it remains to

evaluate the plug flow and well-mixed flow volumes in
the tundish. For this, two approaches, based on elther
the use of the combinedmodel or the use of dlspersion
model,1) are suggested here. Thechoice of the approach
should depend upon the shape of the experimental C-
curve. Ideally, a plug flow and a mixed flow in series give
a C-curve as shownin Fig.